# Toward the profinite rigidity of hyperbolic 3-manifolds

## TL;DR

This paper reviews recent advances in understanding whether hyperbolic 3-manifolds can be uniquely identified by their profinite completions, highlighting progress, evidence, and methods in this area.

## Contribution

It surveys recent results on profinite rigidity of hyperbolic 3-manifolds, reducing the problem to the hyperbolic case and summarizing key techniques used.

## Key findings

- Profinite detection of 3-manifold decompositions
- Profinite rigidity results for certain 3-manifolds
- Evidence supporting hyperbolic case rigidity

## Abstract

This note surveys recent progress toward the profinite rigidity of orientable finite-volume hyperbolic 3-manifolds. Beginning in a brief review of some basic settings of profinite completion and rigidity of general groups, we state the complete results in profinite detection of 3-manifold decompositions and profinite rigidity or nonrigidity of other types of 3-manifolds, which reduce the problem to the hyperbolic case. Then we give some evidence to a positive answer and conclude some existing ideas toward the remained hyperbolic case. We also summarize the methods and techniques used in the process.